The Beeb are not a government organisation. They are independent of HMG (at least in name).
They are not directly tax-funded. Subjects in the UK pay a TV licence (currently about £145, IIRC) that subsidises the organisation in the UK.
The BBC in the UK are, I believe, non-profit - meaning only that they spend all of their £5 billion+ endowment each year. The details are less clear on BBC Worldwide, at least as far as I can find; I have a sneaking suspicion that that organisation is actually a for-profit centre and somewhat independent from the BBC in the UK. Don't quote me on that, however - it's conjecture and speculation on my part.